2. Revisiting some Familiar Concepts
State of a System: The different
configurations in which a system can exist
Specifying the State of a system: State of a
mechanical system is specified by the
positions and momenta of all particles of
the system
3.
Phase space: a space in which all possible
states of a system are represented, with
each possible state of the system
corresponding to one unique point in the
phase space.
•Every state is a point in the
phase space and the dynamics
of a system can be visualised
as its movement through the
phase space.
5. Characteristics of a good Integrator
Minimal need to compute forces(a very
expensive calculation)
Small propagation error, allowing large
time steps
Accuracy
Conserves energy and momentum
Time reversible
Area preserving(Symplectic)
11. Verlet algorithm: Advantages
Integration does not require the velocities,
only position information is taken into
account.
Only a single force evaluation per integration
cycle. (Force evaluation is the most
computationally expensive part in the
simulation).
This formulation, which is based on forward
and backward expansions, is naturally
reversible in time (a property of the
equation of motion).
13. Verlet Algorithm: Disadvantages
Error in velocity approximation is of the
order of time step squared(large errors).
Need to know r(n+1)to calculate v(n).
Numerical imprecision in adding small and
large numbers.
19. Leap Frog : Advantages
Eliminates addition of small numbers to
large ones. Reduces the numerical error
problem of the Verlet algorithm. Here
O(Δt1) terms are added to O(Δt0) terms.
Hence Improved evaluation of velocities.
Direct evaluation of velocities gives a
useful handle for controlling the
temperature in the simulation.
20. Leap Frog : Disadvantages
The velocities at time t are still
approximate.
Computationally a little more expensive
than Verlet.
26. Verlet Algorithm:Advantages
is a second order integration scheme, i. e. the error
term is O ((∂t)3).
is explicit, i. e. without reference into the future:
The system at time t+∂t can be calculated directly
from quantities known at time t.
is self-starting, i. e. without reference into the far
past: The system at time ∂t can be calculated
directly knowing only the system at time t = 0.
allows ∂t to be chosen differently for each time.
This can be very useful when the accelerations vary
strongly over time.
requires only one evaluation of the accelerations
per timestep.
27. Gear Predictor-Corrector
Algorithm
Predict r(t+dt) from the Taylor expansion
at the starting point.
Using this value of r(t+dt), calculate
ac(t+dt).
Similarly calculate v(t+dt), a(t+dt),b(t+dt)
at that point.
The difference between the a(t+dt) and
the predicted ac(t+dt):
31. Gear Predictor-Corrector Algorithm:
Disadvantages
Not time reversible.
Not symplectic, therefore does not
conserve phase space volume.
Not energy conserving, implies over time
there is a gradual energy drift.
